f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

NAG Library Function Documentnag_dgbtrs (f07bec)

1  Purpose

nag_dgbtrs (f07bec) solves a real band system of linear equations with multiple right-hand sides,
 $AX=B or ATX=B ,$
where $A$ has been factorized by nag_dgbtrf (f07bdc).

2  Specification

 #include #include
 void nag_dgbtrs (Nag_OrderType order, Nag_TransType trans, Integer n, Integer kl, Integer ku, Integer nrhs, const double ab[], Integer pdab, const Integer ipiv[], double b[], Integer pdb, NagError *fail)

3  Description

nag_dgbtrs (f07bec) is used to solve a real band system of linear equations $AX=B$ or ${A}^{\mathrm{T}}X=B$, the function must be preceded by a call to nag_dgbtrf (f07bdc) which computes the $LU$ factorization of $A$ as $A=PLU$. The solution is computed by forward and backward substitution.
If ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, the solution is computed by solving $PLY=B$ and then $UX=Y$.
If ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$, the solution is computed by solving ${U}^{\mathrm{T}}Y=B$ and then ${L}^{\mathrm{T}}{P}^{\mathrm{T}}X=Y$.

4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Arguments

1:    $\mathbf{order}$Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 2.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{trans}$Nag_TransTypeInput
On entry: indicates the form of the equations.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$AX=B$ is solved for $X$.
${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$
${A}^{\mathrm{T}}X=B$ is solved for $X$.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, $\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:    $\mathbf{kl}$IntegerInput
On entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
5:    $\mathbf{ku}$IntegerInput
On entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
6:    $\mathbf{nrhs}$IntegerInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs}}\ge 0$.
7:    $\mathbf{ab}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdab}}×{\mathbf{n}}\right)$.
On entry: the $LU$ factorization of $A$, as returned by nag_dgbtrf (f07bdc).
8:    $\mathbf{pdab}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array ab.
Constraint: ${\mathbf{pdab}}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
9:    $\mathbf{ipiv}\left[\mathit{dim}\right]$const IntegerInput
Note: the dimension, dim, of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the pivot indices, as returned by nag_dgbtrf (f07bdc).
10:  $\mathbf{b}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{nrhs}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdb}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $r$ right-hand side matrix $B$.
On exit: the $n$ by $r$ solution matrix $X$.
11:  $\mathbf{pdb}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
12:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{kl}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{kl}}\ge 0$.
On entry, ${\mathbf{ku}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ku}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}>0$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}>0$.
NE_INT_2
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
NE_INT_3
On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$, ${\mathbf{kl}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ku}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7  Accuracy

For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
 $E≤ckεPLU ,$
$c\left(k\right)$ is a modest linear function of $k={k}_{l}+{k}_{u}+1$, and $\epsilon$ is the machine precision. This assumes $k\ll n$.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞ x∞ ≤ckcondA,xε$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$.
Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$, and $\mathrm{cond}\left({A}^{\mathrm{T}}\right)$ can be much larger (or smaller) than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling nag_dgbrfs (f07bhc), and an estimate for ${\kappa }_{\infty }\left(A\right)$ can be obtained by calling nag_dgbcon (f07bgc) with ${\mathbf{norm}}=\mathrm{Nag_InfNorm}$.

8  Parallelism and Performance

nag_dgbtrs (f07bec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dgbtrs (f07bec) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is approximately $2n\left(2{k}_{l}+{k}_{u}\right)r$, assuming $n\gg {k}_{l}$ and $n\gg {k}_{u}$.
This function may be followed by a call to nag_dgbrfs (f07bhc) to refine the solution and return an error estimate.
The complex analogue of this function is nag_zgbtrs (f07bsc).

10  Example

This example solves the system of equations $AX=B$, where
 $A= -0.23 2.54 -3.66 0.00 -6.98 2.46 -2.73 -2.13 0.00 2.56 2.46 4.07 0.00 0.00 -4.78 -3.82 and B= 4.42 -36.01 27.13 -31.67 -6.14 -1.16 10.50 -25.82 .$
Here $A$ is nonsymmetric and is treated as a band matrix, which must first be factorized by nag_dgbtrf (f07bdc).

10.1  Program Text

Program Text (f07bece.c)

10.2  Program Data

Program Data (f07bece.d)

10.3  Program Results

Program Results (f07bece.r)